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| Mirrors > Home > MPE Home > Th. List > psrvscacl | Structured version Visualization version GIF version | ||
| Description: Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| psrvscacl.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | 
| psrvscacl.n | ⊢ · = ( ·𝑠 ‘𝑆) | 
| psrvscacl.k | ⊢ 𝐾 = (Base‘𝑅) | 
| psrvscacl.b | ⊢ 𝐵 = (Base‘𝑆) | 
| psrvscacl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| psrvscacl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) | 
| psrvscacl.y | ⊢ (𝜑 → 𝐹 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| psrvscacl | ⊢ (𝜑 → (𝑋 · 𝐹) ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | psrvscacl.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 2 | psrvscacl.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝑅) | |
| 3 | eqid 2736 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 4 | 2, 3 | ringcl 20248 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) | 
| 5 | 4 | 3expb 1120 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) | 
| 6 | 1, 5 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) | 
| 7 | psrvscacl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
| 8 | fconst6g 6796 | . . . . 5 ⊢ (𝑋 ∈ 𝐾 → ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) | |
| 9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) | 
| 10 | psrvscacl.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
| 11 | eqid 2736 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
| 12 | psrvscacl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
| 13 | psrvscacl.y | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
| 14 | 10, 2, 11, 12, 13 | psrelbas 21955 | . . . 4 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) | 
| 15 | ovex 7465 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
| 16 | 15 | rabex 5338 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V | 
| 17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) | 
| 18 | inidm 4226 | . . . 4 ⊢ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∩ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
| 19 | 6, 9, 14, 17, 17, 18 | off 7716 | . . 3 ⊢ (𝜑 → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘f (.r‘𝑅)𝐹):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) | 
| 20 | 2 | fvexi 6919 | . . . 4 ⊢ 𝐾 ∈ V | 
| 21 | 20, 16 | elmap 8912 | . . 3 ⊢ ((({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘f (.r‘𝑅)𝐹) ∈ (𝐾 ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ↔ (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘f (.r‘𝑅)𝐹):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) | 
| 22 | 19, 21 | sylibr 234 | . 2 ⊢ (𝜑 → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘f (.r‘𝑅)𝐹) ∈ (𝐾 ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) | 
| 23 | psrvscacl.n | . . 3 ⊢ · = ( ·𝑠 ‘𝑆) | |
| 24 | 10, 23, 2, 12, 3, 11, 7, 13 | psrvsca 21970 | . 2 ⊢ (𝜑 → (𝑋 · 𝐹) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘f (.r‘𝑅)𝐹)) | 
| 25 | reldmpsr 21935 | . . . . . 6 ⊢ Rel dom mPwSer | |
| 26 | 25, 10, 12 | elbasov 17255 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) | 
| 27 | 13, 26 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) | 
| 28 | 27 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) | 
| 29 | 10, 2, 11, 12, 28 | psrbas 21954 | . 2 ⊢ (𝜑 → 𝐵 = (𝐾 ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) | 
| 30 | 22, 24, 29 | 3eltr4d 2855 | 1 ⊢ (𝜑 → (𝑋 · 𝐹) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 Vcvv 3479 {csn 4625 × cxp 5682 ◡ccnv 5683 “ cima 5687 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 ↑m cmap 8867 Fincfn 8986 ℕcn 12267 ℕ0cn0 12528 Basecbs 17248 .rcmulr 17299 ·𝑠 cvsca 17302 Ringcrg 20231 mPwSer cmps 21925 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-tset 17317 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mgp 20139 df-ring 20233 df-psr 21930 | 
| This theorem is referenced by: psrlmod 21981 psrass23l 21988 psrass23 21990 mpllsslem 22021 psdvsca 22169 | 
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